Understanding Power Functions: Definitions, Examples, and Practice

1. Section 3.1 Power Functions

2. Objectives: 1. To define and evaluate power functions. 2. To define even and odd functions. 3. To graph power functions and identify the domain and range.

3. Definition Power functionA function of the form f(x) = Cxn where C, n {real numbers}.

4. Notice that the definition includes functions in which n is rational or irrational. We will only be looking at functions with positive integral exponents. This means they will be polynomial functions and require you to use your knowledge of polynomials.

5. Power functions differ from polynomial functions in that they only have one term, and exponents can be any real number. Polynomial functions can have only non-negative integer exponents.

6. EXAMPLE 1For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3). Find the degree, the domain, the range, and graph the function. f(-1) = -2(-1)4 = -2(1) = -2 f(0) = -2(0)4 = -2(0) = 0 f(1/2) = -2(1/2)4 = -2(1/16) = -1/8 f(3) = -2(3)4 = -2(81) = -162

7. EXAMPLE 1For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3). Find the degree, the domain, the range, and graph the function. The degree is 4 D = {real numbers} R = {y|y  0}

8. 2 1 -3 -2 -1 1 2 3 -1 -2 -3 -4 -5 EXAMPLE 1For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3). Find the degree, the domain, the range, and graph the function.

9. 6 4 2 -4 -2 2 4 -2 -4 EXAMPLE 2Graph g(x) = x3. Give the domain and range. D = {real numbers} R = {real numbers}

10. All equation of the form f(x) = Cxn are functions (passing the vertical line test) with domain D = {real numbers}. A parabola also always has line symmetry.

11. Practice: Graph f(x) = -x4. Give the domain and range. 0 = - x4 Roots: x = 0, multiplicity 4 f(1) = - (1)4 = - 1 f(-1) = - (-1)4 = - 1 y-axis symmetry (line symmetry)

12. y x Domain: all real numbers Range: (-, 0]

13. Practice: Graph f(x) = 2x3. Give the domain and range. 0 = 2x3 Roots: x = 0 multiplicity 3 f(1) = 2(1)3= 2 f(-1) = 2(-1)3 = -2 origin symmetry (point symmetry)

14. y x Domain: all real numbers Range: all real numbers

15. Definition Even function A function is even if and only if f(x) = f(-x),  x  Df.

16. Definition Odd function A function is odd if and only if f(-x) = -f(x),  x  Df.

17. Power functions of even degree are even functions and power functions of odd degree are odd functions.

18. One special function is the identity function, y = x. This is a power function of degree 1. The identity function is an odd function.

19. EXAMPLE 3Determine whether the following functions are even, odd, or neither. f(x) = x3 f(-x) = (-x)3 = -x3 -f(x) = -(x3) = -x3 Since f(-x) = -f(x) the function is odd.

20. EXAMPLE 3Determine whether the following functions are even, odd, or neither. g(x) = x4 + x2 g(-x) = (-x)4 + (-x)2 = x4 + x2 -g(x) = -(x4 + x2) = -x4 – x2 Since g(x) = g(-x) the function is even.

21. EXAMPLE 3Determine whether the following functions are even, odd, or neither. h(x) = x2 + 2x + 5 h(-x) = (-x)2 + 2(-x) + 5 = x2 – 2x + 5 -h(x) = -(x2 + 2x + 5) = -x2 – 2x – 5 Since h(x)  h(-x)  -h(x) the function is neither even nor odd.

22. Practice:Classify the function f(x) = 2x3 – 5. 1. Even 2. Odd 3. Neither

23. Practice:Identify the domain of the function f(x) = 2x3 – 5. 1. {x|x  real numbers} 2. {y|y  -5} 3. {y|y  real numbers} 4. None of these

24. Practice: Determine whether the following functions are even, odd, or neither. f(x) = 4x5 + 2x3 – x f(-x) = - 4x5 - 2x3 + x -f(x) = -(4x5 + 2x3 - x) = - 4x5 - 2x3 + x f(-x)= - f(x)  f(x) is odd.

25. Practice: Determine whether the following functions are even, odd, or neither. g(x) = 3x4 - 5x2 g(-x) = 3(-x)4 - 5(-x)2 g(-x) = 3x4 - 5x2 g(x) = g(-x) Therefore, g(x) is even.

26. Practice: Determine whether the following functions are even, odd, or neither. h(x) = x3 - x2 + x - 1 h(-x) = (-x)3 - (-x)2 + (- x) - 1 h(-x) = - x3 - x2 - x -1 -h(x) = - (x3 - x2 + x - 1) -h(x) = - x3 + x2 - x + 1 h(x)  h(-x) h(-x)  -h(x) Therefore, h(x) is neither even nor odd.

27. Homework: pp. 109-110

28. ►A. Exercises Graph each power function. Give the domain and range of each and classify as even or odd. 5. f(x) = -1/4x4

29. ►A. Exercises Graph each power function. Give the domain and range of each and classify as even or odd. 7. y = 5/12x24

30. ►B. Exercises Evaluate. 13. f(-3) for f(x) = -2x3

31. ►B. Exercises Evaluate. 15. f(-17.95) for f(x) = -2.5x16

32. ►B. Exercises For f(x) = Cxn. 16. Find f(1)

33. ►B. Exercises For f(x) = Cxn. 17. Find f( 3) n

34. ►B. Exercises For f(x) = Cxn. 18. Find all zeros.

35. ►B. Exercises For f(x) = Cxn. 19. What is the multiplicity of the zero?

36. ►B. Exercises For f(x) = Cxn. 20. Give the domain of f(x).

37. ►B. Exercises For f(x) = Cxn. 21. Give the range of f(x) if n is odd.

38. ►B. Exercises For f(x) = Cxn. 22. If n is even, on what does the range of f(x) depend?

39. ►B. Exercises For f(x) = Cxn. 23. Give the range of f(x) if n is even.

40. ■ Cumulative Review 28. In ABC, find C, given a = 47, b = 63, and c = 82.

41. ■ Cumulative Review 29. Is the relation in the graph a function?

42. ■ Cumulative Review 30. If sin x = 0.3, find csc x, cos (90 – x), and sec (90 – x).

43. ■ Cumulative Review 31. Solve 2x² – 5x + 7 = x(x + 1).

44. ■ Cumulative Review 32. Find the slope of the line joining (2, 7) to (4, -5).